An axiomatization of Herzberger's $2$-dimensional presuppositional semantics.
John N. T. Martin · Notre Dame Journal of Formal Logic · 1977
The purpose of this paper* is to axiomatize two 4-valued propositional logics suggested by Herzberger in [l], section VI.They are of philosophical interest because their interpretation makes use of two ideas inspired by Jean Buridan: (1) a proposition may correspond to the world and yet be untrue because it is semantically deviant, and (2) logically valid arguments preserve correspondence with reality, not truth.If the two non-classical truth-values of these systems are identified, the resulting tables for the classical connectives are the weak and strong systems of Kleene.Unlike Kleene's system, the 4-valued ones offer a choice of designated values that renders semantic entailment perfectly classical.Compare Herzberger [2] and Martin [5].Let the set 9r of formulas be inductively defined over a denumerable set of atomic formulas such that Ί A, A & B, CA, BA, TA f FA, XA, and fA are formulas if A and B are.Let W be the set of all m such that for some v and to, (1) for any atomic formula A, v(A), to(A) e {θ, l};0 otherwise; *(CA) = n(Bi4) = lι(TA) = ti(FA) = n(tA) = υ(fA) = 1; *I would like to thank Leo Simons for his helpful comments on a draft of this paper.